Average Error: 10.7 → 1.3
Time: 4.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r512245 = x;
        double r512246 = y;
        double r512247 = z;
        double r512248 = t;
        double r512249 = r512247 - r512248;
        double r512250 = r512246 * r512249;
        double r512251 = a;
        double r512252 = r512251 - r512248;
        double r512253 = r512250 / r512252;
        double r512254 = r512245 + r512253;
        return r512254;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r512255 = z;
        double r512256 = t;
        double r512257 = r512255 - r512256;
        double r512258 = a;
        double r512259 = r512258 - r512256;
        double r512260 = r512257 / r512259;
        double r512261 = y;
        double r512262 = x;
        double r512263 = fma(r512260, r512261, r512262);
        return r512263;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.7
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 10.7

    \[x + \frac{y \cdot \left(z - t\right)}{a - t}\]

  2. Simplified3.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)}\]
  3. Using strategy rm

  4. Applied clear-num3.2

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, z - t, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef3.2

    \[\leadsto \color{blue}{\frac{1}{\frac{a - t}{y}} \cdot \left(z - t\right) + x}\]

  7. Simplified2.9

    \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y}}} + x\]
  8. Using strategy rm

  9. Applied *-un-lft-identity2.9

    \[\leadsto \frac{z - t}{\frac{a - t}{y}} + \color{blue}{1 \cdot x}\]

  10. Applied *-un-lft-identity2.9

    \[\leadsto \color{blue}{1 \cdot \frac{z - t}{\frac{a - t}{y}}} + 1 \cdot x\]

  11. Applied distribute-lft-out2.9

    \[\leadsto \color{blue}{1 \cdot \left(\frac{z - t}{\frac{a - t}{y}} + x\right)}\]

  12. Simplified1.3

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)}\]

  13. Final simplification1.3

    \[\leadsto \mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

  (+ x (/ (* y (- z t)) (- a t))))